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A cube contains 9 identical spheres, as shown below. There is one sphere in the center of the cube. Above (and below) the center sphere are 4 spheres tangent to the center sphere and the 4 corners of the cube. What is the radius of each sphere?
As usual, watch the video for a solution.
Or keep reading.
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“All will be well if you use your mind for your decisions, and mind only your decisions.” Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.
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Answer To Spheres In A Cube Puzzle
(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).
Suppose a cube has edges of side length s. Then each face has a diagonal equal to s√2. Then the diagonal of the cube is the hypotenuse of a right triangle with legs consisting of one edge and one face diagonal. Thus the cube has a diagonal in length equal to:
√(s2 + (s√2)2)
= s√3
In our problem s = 10 so the diagonal is 10√3.
Now let’s consider the 3 spheres along the diagonal of the cube. There is a cube formed between the lower left sphere and the corner of the cube with an edge length equal to r. Thus the distance between the center of that sphere and the corner of the cube is r√3. This is true for the other corner of the sphere. In between those two spheres are lengths of r, 2r, and r:
Thus the diagonal of the cube has a length equal to r(4 + 2√3), and this equals 10√3, so we have:
r(4 + 2√3) = 10√3
r = 10√3/(4 + 2√3)
r = 10√3/(4 + 2√3) [(4 – 2√3)/(4 – 2√3)]
r = (40√3 – 60)/(16 – 12)
r = (40√3 – 60)/4
r = 10√3 – 15 ≈ 2.32
Reference
Puzzling StackExchange post which credits the UK National Mathematics Contest
http://puzzling.stackexchange.com/questions/41952/nine-identical-spheres-fit-exactly-into-a-cube
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